On the Riemann-Siegel formula
نویسنده
چکیده
In this article we derive a generalization of the Riemann-Siegel asymptotic formula for the Riemann zeta function. By subtracting the singularities closest to the critical point we obtain a significant reduction of the error term at the expense of a few evaluations of the error function. We illustrate the efficiency of this method by comparing it to the classical Riemann-Siegel formula.
منابع مشابه
High precision computation of Riemann's zeta function by the Riemann-Siegel formula, I
We present rigorous and sharp bounds for the terms and remainder in the Riemann-Siegel formula (for a general argument, not necessarily on the critical line). This allows for the computation of ζ(s) and Z(t) to high precision. We also derive the Riemann-Siegel formula in a new and more direct way.
متن کاملRiemann’s Saddle-point Method and the Riemann-Siegel Formula
Riemann’s way to calculate his zeta function on the critical line was based on an application of his saddle-point technique for approximating integrals that seems astonishing even today. His contour integral for the remainder in the Dirichlet series for the zeta function involved not an isolated saddle, nor a saddle near a pole or an end-point or several coalescing saddles, but the configuratio...
متن کاملThe Development of a Hybrid Asymptotic Expansion for the Hardy Function , Consisting of Just Main Sum Terms, Some Less than the Celebrated Riemann-siegel Formula
This paper begins with a re-examination of the Riemann-Siegel Integral, which first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L. Siegel on his 1932 account of Riemann’s unpublished work on the zeta function. By application of standard asymptotic methods for integral estimation, and the use of certain approximations pertaining to special functions, it proves possible to ...
متن کاملTheta and Riemann xi function representations from harmonic oscillator eigensolutions
From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A key result provides a basis for generalizing the important Riemann-Siegel integral formula.
متن کاملSiegel-Veech constants for strata of moduli spaces of quadratic differentials
We present an explicit formula relating volumes of strata of meromorphic quadratic differentials with at most simple poles on Riemann surfaces and counting functions of the number of flat cylinders filled by closed geodesics in associated flat metric with singularities. This generalizes the result of Athreya, Eskin and Zorich in genus 0 to higher genera.
متن کامل